一类推广的Hilbert型不等式

A Class of Generalized Hilbert-Type Inequalities

Hilbert型不等式是解析不等式的重要组成部分,在分析学以及相关领域有着极为重要的作用。通过引入若干参数,构造了一个一般形态的分式型核函数,并利用权系数的方法和实分析的技巧,建立了含有最佳常数因子的Hilbert型不等式,推广了相关文献的结果。此外,借助余切函数的部分分式展开公式,给出了所构建不等式的最佳常数因子的三角函数表示形式。

Hilbert-type inequality is an important part of analytic inequality,and plays a very important role in analysis and related fields.By introducing some parameters,a fractional kernel function in general form is constructed.Using the method of weight coefficient and the technique of real analysis,a Hilbert-type inequality with the best constant factor is established,which generalizes the results of relevant literature.In addition,with the help of the partial fraction expansion formula of the cotangent function,the trigonometric function representation of the best constant factor in the newly obtained inequality is given.